Final answer:
Using the Pythagorean theorem, after traveling 605 miles due north, the barge captain has approximately 981 miles left to travel due west.
Step-by-step explanation:
The student is tasked with a navigation problem, which involves calculating the distance left for the barge to travel. Given that the barge has to travel in two legs, one due north and one due west, and that the direct route's distance is 1,152 miles, we can use the Pythagorean theorem to solve this.
After traveling 605 miles due north, we can treat this problem as a right-angled triangle where 605 miles is one leg and we have to find the length of the other leg (due west).
Using the Pythagorean theorem:
c² = a² + b²
where c is the hypotenuse (1,152 miles) and a is one leg (605 miles). We solve for b (the distance yet to be traveled due west).
b² = c² - a²
b² = 1,152² - 605²
b² = 1,327,904 - 366,025
b² = 961,879
b = √961,879
Rounding to the nearest mile, b ≈ 981 miles.
Therefore, the barge captain has approximately 981 miles left to travel due west to reach her destination.